Seminario di Calcolo delle Variazioni & Equazioni alle Derivate Parziali

Starting from classical examples of variational inequalities, we deal with recent regularity results for a class of obstacle problems under standard and nonstandard growth conditions. Our approach goes into the direction of Hoelder regularity, starting from the pioneering work of E. De Giorgi. We conclude with some open questions concerning regularity of the free boundary in the variable exponent setting

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Abstract: Viene proposto un modello matematico nonlineare che evidenzia un accoppiamento tra diversi modi di vibrazioni in un ponte sospeso. Con un opportuno metodo di Galerkin vengono stimate le soglie di stabilità strutturale. Si ottengono sia stime teoriche (basate su criteri di stabilità per le equazioni di Hill), sia stime numeriche. Viene così evidenziata l'esistenza di una soglia critica di energia oltre la quale le oscillazioni verticali si possono trasformare in oscillazioni torsionali.

Abstract: We prove the precise decay in time estimates of the total mass of a solution to the Cauchy problem for quasilinear degenerate parabolic equation with absorption terms. The equation may involves additional terms which behavior at in?nity are described by Wolf potentials. In terms of new critical exponents we control precise behavior of a total mass.

Abstract: The Lipschitz truncation method allows to approximate a Sobolev function by Lipschitz functions with only minor modifications of the original function. This technique is closely related to the Calderon-Zygmund decomposition of Sobolev function. The Lipschitz truncation can for example be used to identify the non-linear stress for Non-Newtonian fluids although only weak converge is at hand. In the time-dependent case the problem becomes much more difficult due to the presence of the pressure. This problem can be solved by introducing a solenoidal Lipschitz truncation. Another application of the Lipschitz is a direct proof of the so called harmonic and caloric approximation lemmas.

Abstract: The Plateau's problem, named after the Belgian physicist J. Plateau, is a classic in the calculus of variations and regards minimizing the area among all surfaces spanning a given contour. Although Plateau's original concern were $2$-dimensional surfaces in the $3$-dimensional space, generations of mathematicians have considered such problem in its generality. A successful existence theory, that of integral currents, was developed by De Giorgi in the case of hypersurfaces in the fifties and by Federer and Fleming in the general case in the sixties. When dealing with hypersurfaces, the minimizers found in this way are rather regular: the corresponding regularity theory has been the achievement of several mathematicians in the 60es, 70es and 80es (De Giorgi, Fleming, Almgren, Simons, Bombieri, Giusti, Simon among others). In codimension higher than one, a phenomenon which is absent for hypersurfaces, namely that of branching, causes very serious problems: a famous theorem of Wirtinger and Federer shows that any holomorphic subvariety in $\mathbb C^n$ is indeed an area-minimizing current. A celebrated monograph of Almgren solved the issue at the beginning of the 80es, proving that the singular set of a general area-minimizing (integral) current has (real) codimension at least 2. However, his original (typewritten) manuscript was more than 1700 pages long. In a recent series of works with Emanuele Spadaro we have given a substantially shorter and simpler version of Almgren's theory, building upon large portions of his program but also bringing some new ideas from partial differential equations, metric analysis and metric geometry.

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Abstract: We propose to describe plane flow of a conducting fluid in a thin layer between two ferromagnetic plates. The model leads to a coupled system of two PDEs in 2D, namely the scalar Maxwell equation for the magnetic field which is assumed perpendicular to the plane of motion, and the vector Navier-Stokes equation for the fluid velocity. Both equations contain hysteresis terms which are due to the ferromagnetic reaction of the surrounding medium. To ensure thermodynamic consistency of the model, hysteresis is expressed in terms of the Preisach model. The main result includes existence and uniqueness of a strong solution for regular data as long as the magnetic field stays in the convexity domain of the Preisach operator (joint work with M. Eleuteri and J. Kopfova).

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Abstract: In this talk we will present qualitative and numerical results for a partial differential equation (PDE) system which models a fluid-structure PDE of longstanding interest within the mathematical literature. The coupled PDE model under discussion involves a Stokes or Navier-Stokes system, which evolves on a three dimensional domain, interacting with a fourth order plate equation which evolves on a flat portion of said fluid domain. Among other technical difficuties we note that, inasmuch as the fluid velocity does not vanish on all of the boundary, the associated pressure variable cannot be eliminated via the classic Leray Projector. We will discuss how wellposedness of this fluid-structure dynamics is eventually attained via a certain variational "inf-sup" (or Babuska-Brezzi) formulation. Subsequently, we will show how our constructive proof of wellposedness naturally gives rise to a finite element method for numerically approximating solutions to the fluid-structure dynamics. Time permitting, we will also discuss a result of backward uniqueness for this PDE system. (This is joint work with Tom Clark.)

Abstract: We review local regularity in Lebesgue spaces for solutions u of quasilinear elliptic systems. We discuss Meier's result in such a particular case and we show how to improve it, if off diagonal coefficients are small when |u| is large: the faster off diagonal coefficients decay, the higher the integrability of u becomes.

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Abtsract: Abstract: In a seminal paper by Chlebìk, Cianchi and Fusco (Ann. Math. 2005), a sufficient condition for rigidity of equality cases in Steiner inequality was presented. (By rigidity of equality cases we mean the situation when all sets realizing equality in the Steiner's inequality defined by a given symmetric set are necessarily symmetric.) Their condition is however far from being necessary, and the problem of characterizing rigidity of equality cases was left open even when the given symmetric set is a polyhedron. In this talk we first introduce a measure-theoretic notion of connectedness, inspired by Federer's notion of indecomposable current, that is then exploited to prove several characterization results of rigidity of equality cases, both in the case of the classical Steiner's inequality, as well as in the case of Ehrhard's inequality for Gaussian perimeter. This is a joint work with Filippo Cagnetti (U. Sussex), Maria Colombo (SNS Pisa), and Guido De Philippis (U. Bonn).

Abstract: The aim of this talk is to present properties of sections of convex bodies with respect to different types of measures. We will remind a formula connecting the Minkowski functional of a convex symmetric body K with the measure of its sections. We apply this formula to study properties of general measures most of which were known before only in the case of the standard Lebesgue measure. In particular we will present a version of the Isomorphic Busemann-Petty problem for arbitrary measures and its connections to the recent results of Alexander Koldobsky on the slicing inequality for arbitrary measures.